By William L. Briggs
A Multigrid instructional is concise, attractive, and obviously written. Steve McCormick is the one man i do know which could pull off instructing in spandex. simply ensure you take a seat within the again row.
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Additional resources for A Multigrid Tutorial
Think for a moment about the step in the correction scheme that requires transferring the error approximation e2h from the coarse grid 2h to the fine grid h. This is a common procedure in numerical analysis and is generally called interpolation or prolongation. Many interpolation methods could be used. Fortunately, for most multigrid purposes, the simplest of these is quite effective. For this reason, we consider only linear interpolation. The linear interpolation operator will be denoted . 2 shows graphically the action of .
Use the fact that the eigenvalues are given by the Rayleigh quotients of the eigenvectors, = (Awfc, Wfc)/(wfc, Wfc), where wk is the eigenvector associated with 16. Properties of Gauss-Seidel. Assume A is symmetric, positive definite. (a) Show that the jth step of a single sweep of the Gauss-Seidel method applied to Au = f may be expressed as (b) Show that the jth step of a single sweep of the Gauss-Seidel method can be expressed in vector form as where is the jth unit vector. (c) Show that each sweep of Gauss-Seidel decreases the quantity (Ae,e), where e = u — v.
General stationary linear iteration. It was shown that a general stationary linear iteration can be expressed in the form (a) Show that m sweeps of the iteration has the form Find an expression for C(f). (b) Show that the form of the iteration given above is equivalent to where r(0) is the initial residual. Use this form to argue that the exact solution to the linear system, u, is unchanged by (and is therefore a fixed point of) the iteration. 5. Interpreting Gauss-Seidel. Show that the Gauss-Seidel iteration is equivalent to successively setting each component of the residual to zero.
A Multigrid Tutorial by William L. Briggs